from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([20,21]))
pari: [g,chi] = znchar(Mod(403,1089))
Basic properties
| Modulus: | \(1089\) | |
| Conductor: | \(99\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{99}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1089.s
\(\chi_{1089}(40,\cdot)\) \(\chi_{1089}(94,\cdot)\) \(\chi_{1089}(112,\cdot)\) \(\chi_{1089}(403,\cdot)\) \(\chi_{1089}(457,\cdot)\) \(\chi_{1089}(475,\cdot)\) \(\chi_{1089}(481,\cdot)\) \(\chi_{1089}(844,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.0.159386923550435671074967363509984324121230045171.1 |
Values on generators
\((848,244)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1089 }(403, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)